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The problem involves solving the inequality \( f(x) = \frac{x - 4}{x^2 - 7x - 8} < 0 \).

To solve this rational inequality, let's proceed step-by-step:

1. **Factor the denominator** to find any restrictions and identify critical points. 
   
   The denominator \( x^2 - 7x - 8 \) can be factored as:
   \[
   x^2 - 7x - 8 = (x - 8)(x + 1)
   \]
   So, \( f(x) = \frac{x - 4}{(x - 8)(x + 1)} \).

2. **Identify critical points** by setting the numerator and denominator equal to zero:
   - **Numerator**: \( x - 4 = 0 \Rightarrow x = 4 \)
   - **Denominator**: \( x - 8 = 0 \Rightarrow x = 8 \)
   - **Denominator**: \( x + 1 = 0 \Rightarrow x = -1 \)

   Thus, the critical points are \( x = -1 \), \( x = 4 \), and \( x = 8 \). These points divide the number line into intervals.

3. **Determine the sign of \( f(x) \) in each interval** by testing values within each interval. The intervals are:
   - \( (-\infty, -1) \)
   - \( (-1, 4) \)
   - \( (4, 8) \)
   - \( (8, \infty) \)

4. **Test each interval** to determine where \( f(x) < 0 \):
   - For \( x \in (-\infty, -1) \): Choose \( x = -2 \)
     \[
     f(-2) = \frac{-2 - 4}{(-2 - 8)(-2 + 1)} = \frac{-6}{(-10)(-1)} = \frac{-6}{10} < 0
     \]
     This interval satisfies \( f(x) < 0 \).
   
   - For \( x \in (-1, 4) \): Choose \( x = 0 \)
     \[
     f(0) = \frac{0 - 4}{(0 - 8)(0 + 1)} = \frac{-4}{-8} = \frac{1}{2} > 0
     \]
     This interval does not satisfy \( f(x) < 0 \).

   - For \( x \in (4, 8) \): Choose \( x = 5 \)
     \[
     f(5) = \frac{5 - 4}{(5 - 8)(5 + 1)} = \frac{1}{-3 \cdot 6} = \frac{1}{-18} < 0
     \]
     This interval satisfies \( f(x) < 0 \).

   - For \( x \in (8, \infty) \): Choose \( x = 9 \)
     \[
     f(9) = \frac{9 - 4}{(9 - 8)(9 + 1)} = \frac{5}{1 \cdot 10} = \frac{1}{2} > 0
     \]
     This interval does not satisfy \( f(x) < 0 \).

5. **Write the solution**: The inequality \( f(x) < 0 \) holds for \( x \in (-\infty, -1) \cup (4, 8

Solve the inequality \( f(x) = \frac{x - 4}{x^2 - 7x - 8} < 0 \) and provide the solution in interval notation.

This guide provides a step-by-step solution to the rational inequality \( \frac{x - 4}{x^2 - 7x - 8} < 0 \). Learn about factoring quadratics, identifying critical points, and testing intervals to determine the solution in interval notation. Suitable for college algebra students and advanced high school learners.

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